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13 Fundamental of Spectroscopy for Optical Remote Sensing Xinzhao Chu I Principle of Uncertainty Indeterminacy 0. Expression of Heisenberg s Principle of Uncertainty It is worth to point out that the principle of uncertainty originates from the wave-particle duality of matter. It is a nature s law, not due to measurement technologies. 1. Interpretation of Uncertainty Principle: It is impossible to determine the position and momentum of a particle to an arbitrary degree of accuracy. Examples to show the effects of h 0 in macro-world and in micro-world And demonstrate why a motion orbit is meaningful only in macro-world. 2. Derivation of Uncertainty Principle from Commutation Relation This is a good reference for students but not to be discussed in class (due to time constraint). This derivation shows that the uncertainty relation can be derived exactly from quantum mechanics commutation relation. The principle of uncertainty is fundamental to quantum mechanics, but not an approximation. 3. Generalization of Heisenberg s Principle of Uncertainty ΔA ΔB 1 2 < [ ˆ A, ˆ B ] > Derive ΔE Δt 2 from Δp Δx 2 4. The Intrinsic Nature of Uncertainty Principle is the Wave-Particle Duality of Matter (all material particles, photons and electromagnetic waves). It is a law of the nature, not due to measurement technologies or imperfection. Give an example of electron diffraction to demonstrate 1) Why the uncertainty is on the order of h Planck constant. 2) The unequal fringe heights in the double-slit experiment are due to the modulation of interference fringes by diffraction. 3) Diffraction s nature is the multiple-beam interference. 5. Applications of Principle of Uncertainty This is to demonstrate the values of this principle in solving real problems, in which experimental facts cannot be explained well by classical physics. 1) The minimum kinetic energy of bound particle 2) Electrons inside an atom cannot fall in the nucleus 3) Natural linewidth of spectral lines

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19 Fundamental of Spectroscopy for Optical Remote Sensing Xinzhao Chu I The Intrinsic Nature of Uncertainty Principle is the Wave-Particle Duality of Matter (all material particles, photons and electromagnetic waves). It is a law of the nature, not due to measurement technologies or imperfection. An Example for Principle of Uncertainty Electron Diffraction Diagram of electron diffraction (taken from Professor Chu s Atomic Physics) Let the slit width be Δq. Since we cannot determine by which point of the slit the electron goes through, the position uncertainty of the electron is Δq. Electron generates a diffraction pattern on the screen, so its motion direction can deviate from the original by angle of α. Thus, the momentum uncertainty Δp = p sinα pα. Let us use the first dark fringe position as a reference α o λ/δq, then we have Δp pα o = pλ/δq. From de Broglie relationship, we know λ = h/p. By substituting this de Broglie wavelength into Δp equation, we get Δp pλ/δq = h/δq. Therefore, we have Δp Δq ~ h. This is larger than 2 = h, which is allowed by the uncertainty principle. 4π Note: 1) The unequal fringe heights in the double-slit experiment are due to diffraction modulating the interference fringes. 2) Diffraction s nature is multiple-beam interference, as shown in the diagram below. Nature of diffraction is multiple-beam interference (from Professor Zhao s Optics)

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21 Fundamental of Spectroscopy for Optical Remote Sensing Xinzhao Chu I Application Examples of Uncertainty Principle (taken from Yang s Atomic Physics) Example 1. The minimum mean kinetic energy of a bound particle Assume a particle is confined within a range of r, so its position uncertainty is Δx = r. According to the principle of uncertainty, its momentum uncertainty has a minimum of Δp x = 2Δx = 2r Recall the definition of Δp x is Δp x = < ( p x p x ) 2 >. For a particle bound in certain space, its momentum in any direction should have a mean of zero, i.e., p x = 0. Therefore, we have In a 3-dimentional space, (Δp x ) 2 =< p x 2 >. < p x 2 >= 1 3 < p2 >. Therefore, in non-relativity range, the minimum mean kinetic energy is given by E k = < p2 > 2m = 3 2 8mr 2. It is obvious from this equation that the particle s kinetic energy cannot be zero! This conclusion does NOT depend on how the particle is confined. As long as a particle is in a potential well, its minimum kinetic energy cannot be zero. That is, the particle cannot fall to the bottom of the potential well. In fact, if the particle s kinetic energy is zero, then the principle of uncertainty tells us that its Δx -- how can the particle be confined?! Example 2. Why can t the electrons inside an atom fall in the nucleus? Bohr s theory cannot explain why the accelerated electron inside a hydrogen atom would not radiate electromagnetic waves (so losing energy) and would not fall in the nucleus. This question is answered by the principle of uncertainty in the following. The electron inside an atom is bound spatially, so its kinetic energy can be given by the equation in Example 1. When the electron is getting closer to the nucleus, that is, r is getting smaller, Δx changes from the atomic dimension of 0.1 nm to the nuclear size of fm (10-15 m). According to the principle of uncertainty, the electron s momentum uncertainty increases, E k = < p2 > 2m = 3 2 8mr 2 or using the equation in Example 1, its mean kinetic energy increases significantly. For example, when the electron motion range is decreased from 0.1 nm to 3 fm, its mean kinetic energy is increased from 1 ev to 0.1 GeV. Nowhere can the electron get such large energy; therefore, it is impossible for the electron to even get close to the nucleus, no mention fall inside the nucleus. A nucleus can have protons and neutrons, but not electrons, because the principle of uncertainty does not allow the confinement of electron within a nuclear size.

22 Fundamental of Spectroscopy for Optical Remote Sensing Xinzhao Chu I 13 Example 3. Natural linewidth of a spectral line: Excited state lifetime vs. energy level width From Bohr s theory, we know that when two energy levels corresponding to a spectral line have certain energy values, the transition between these two levels will give a spectral line whose width is infinitely narrow (just like a δ function), according to Bohr s frequency condition hν = E 2 E 1. However, as long as an electron transitions from the upper level to the lower level, the upper energy level must have finite lifetime. Therefore, Δt is not infinite. According to the principle of uncertainty, ΔE 0, i.e., the upper energy level must have certain width ΔE. Consequently, the spectral line cannot be a δ function, but must have certain linewidth. This is called the natural linewidth of the spectral line. For example, if an excited state has a lifetime of Δt = 10-8 s, then ΔE 2Δt = ev. The spectral linewidth can be derived as Δν n = ΔE h 2hΔt = 1 2 2π Δt. Natural linewidth calculation: In reality (taken from Professors Chu and Papen lidar book chapter), the natural linewidth is given by, for Na D 2 line (3 2 P 3/2 3 2 S 1/2 ), τ = ns Δν n = 1 2π τ = 1 2π 16.23ns = 9.8MHz For a metastable state, the lifetime is much longer, e.g., on the order of µs, ms, or s. Then its natural linewidth can be significantly narrower than MHz.

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24 Fundamental of Spectroscopy for Optical Remote Sensing Xinzhao Chu I Equation of Motion Schrödinger Equation 1. Introduction to the Principle of Motion the Schrödinger equation describes the time evolution of the state of a quantum system. 1) Schrödinger equation is a fundamental equation in QM. In principle, it cannot be derived or proven from more fundamental principles, but it can only be verified by experiments. 2) Hamiltonian operator 3) Schrödinger equation in the abstract state space as well as in any representation. 4) There is no indeterminacy in the time evolution of a quantum system. Indeterminacy appears only when a physical quantity is measured, the state vector undergoes an unpredictable modification, i.e., the reduction or collapse of the state to an eigenstate. However, between two measurements, the state vector evolves in a perfectly deterministic way, in accordance with the Schrödinger equation. 5) The Schrödinger equation is linear, so its solution is linearly superposable. 6) Time evolution does not modify the global probability of finding the particle in all space, which always remains Stationary-State Schrödinger Equation Energy Eigenvalue Equation 3. Application Examples of Schrödinger Equation in Solving Atomic Particle Problems Three examples are presented to demonstrate 1) How to solve stationary-state Schrodinger equations 2) How Schrodinger equation naturally leads to quantized energy, etc. 3) Real applications of quantum mechanics in modern physics and technologies Example 1. 1-D Infinite Potential Well Points: Boundary conditions naturally lead to the quantized energy levels and the minimum energy isn t zero; wave function represents a standing wave; the distribution of probability of particle isn t homogeneous Applications: Hydrogen atom s electron is similar to be in quantum well; quantum well in laser, photo detector, etc. Example 2. Potential Barrier Penetration (Finite Potential Well) quantum tunnel effect Points: Particles have probability appearing outside classical physics barriers Applications: emission of α particle by radio-active nucleus, tunnel diode, electron tunneling microscope Example 3. 1-D Harmonic Oscillator Points: Quantized and non-zero energy of oscillator Applications: Quantized radiation field

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